The utilization of Kubelka-Munk (K-M) theory and variations thereof in the paper coating industry for determining scattering coefficients is well known. The K-M theory, along with pore structure information (e.g., pore volume, pore diameters, etc.) obtained from mercury intrusion porosimetry measurements, is widely used to characterize paper coatings and to design, predict or enhance the understanding of coated paper properties.
The principal advantage of using the K-M theory is its mathematical simplicity, but this simplicity is in effect a tradeoff for a loss of explicit connection with the optical properties of the individual components within a diffusive layer that are part of a coating system. One particular problem with the use of the K-M theory as an aid to assessing performance of coating systems is its treatment of scattering as being caused by a homogenous medium, rather than a collection of individual scattering centers. Under the K-M approach, isotropic scattering of light is considered to occur, but in real systems, light is scattered anisotropically.
Another problem with the K-M theory is its limited number of degrees of freedom and the inability to address all of the other variables that exist in a pigmented coating system, e.g., coating thickness, properties of the base sheet, certain properties of the pigment, both physical and optical, the boundary layer conditions associated with the coating layer, etc. Thus, the K-M approach, while sometimes a reasonable approximation depending on the homogeneity of the medium, is deficient in predicting the actual light scattering effects in non-homogenous coating systems having multiple scattering centers as well as boundary layer effects and can therefore only predict performance properties to a certain extent.
Accordingly, a practical need exists to provide improved ways to predict coating performance for paper coating systems, including taking into account the boundary conditions for the coating layer, such as the surface roughness of the coating and the properties of the interfacial region between the base sheet and the coating, coating thickness, anisotropic scattering of light, use of pigments having varied particle size distribution characteristics, and the like. The present invention solves this need through the utilization of a diffusion approximation model and its adaptation through the identification of various inputs to the model that relate to both the physical characteristics and the optical properties of coating systems. Using selected inputs and the diffusion approximation model, one can predict the performance of a coating system. This ability is especially useful when making predictions for coating systems being designed to meet certain targeted optical or physical characteristics for a particular end-use application. In this mode, the model can determine what physical characteristics are needed to meet specific coating performance targets in the form of optical properties such as brightness, gloss, and opacity.
U.S. Pat. No. 6,064,487 to Kettler et al. discloses a process for calculating a color formulation of a pigmented special effect shade that include determining optical reflection factors using a radiation transport model, particularly an azimuth-independent form thereof. Kettler et al. are primarily concerned with developing a paint formulation to match that of an existing automobile. However, the method disclosed in this patent is totally unrelated to modeling the coating systems and coating performance properties that are relevant to the paper industry.